Algebra 1 Unit 8 Interactive Notebooks: Forms of Quadratic Functions

This year I’ve committed to posting each unit of both my Algebra 1 and Algebra 2 INBs.

My district is moving to a standards based curriculum, and has identified priority standards for every course. These are the standards we are required to address and assess our students over, so they pretty much form our units.

You can find my Algebra 1 (year long class) INB posts here:

Unit 1 | Unit 2 | Unit 3 | Unit 4 | Unit 5 | Unit 6 | Unit 7

And my Algebra 2 INB posts here:

Unit 1 | Unit 2 | Unit 3 | Unit 4 | Unit 5 | Unit 6 | Unit 7 | Unit 8

And finally, my posts from a 2nd go around I’m teaching of Algebra 1 here:

Unit 1 | Unit 2 | Unit 3 | Unit 4

 

The 7th priority standard we have in Algebra 1 is F.IF.8:

Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

I think the last three units of our Algebra 1 year tell a great story. They cycle back into each other, each one pushing a little further, and it all comes together (in the next unit!). This unit feeds off of a lot of the solution methods from the last unit, because we utilize some of these methods to rewrite a quadratic in another form.

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Skill 1: I can rewrite a standard form quadratic function in a form that reveals the zeros

We begin this unit by introducing the three forms of quadratic functions and what information they can reveal to us. We won’t actually be graphing them until the next unit, but there needs to be a goal – why do we want these quadratics in different forms? So I introduced the parabola, and some of the vocabulary surrounding it.

Rewriting a standard form quadratic in factored form is pretty much the same as solving a quadratic by factoring, so I just took the chance to address some of the most common mistakes students had been making (not factoring out a GCF, misplacing negatives, getting confused by positive or negative factor pairs) and to go over the process again.

Skill 2: I can rewrite a standard form quadratic function in a form that reveals the vertex

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Again, this process is fairly similar to solving a quadratic by completing the square, so I again took the chance to address some common misconceptions, and we emphasized throughout that our goal was to end up with something that looked like vertex form, so the students came up with a lot of their own ideas about what to do after they completed the square.

Skill 3: I can rewrite a factored form quadratic function in a form that reveals the y-intercept

 

The notes page for this is pretty basic – but I tried to emphasize as we went through it what we were actually doing. What operations are implied in the factored form function? If there’s multiplication implied, how do we multiply the two factors? What should I do with that extra coefficient?

At this point, I wanted to bring in the idea that we need to be able to 1. recognize what form a function is in by looking at it and 2. identify what form we are being asked to rewrite it in. So the last page of these notes throws in a standard form to vertex form problem, and then asks students to identify forms of some other functions and to make observations about how they can quickly identify the form (it’s all about the parentheses, right?)

It was also revealed to me that I apparently don’t know what half of 7.5 is…so just ignore all of those scribbles on that problem where I had to fix my mistake…

Skill 4: I can rewrite a vertex form quadratic function in a form that reveals the y-intercept

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Something that we really focused on when looking at this skill was identifying when you were “done”. Right after you multiply out the squared binomial, it looks like you have standard form already. But, you haven’t used all of the components of the original function! So we made a point of constantly checking back in with the original function – did we use all the parts of it? If not, what parts have we not used and what should we use next in our rewriting?

Skill 5: I can rewrite a quadratic to reveal any key feature

The triangle on the front of this proved very useful as a summary of the unit – I saw students looking at it constantly as they reviewed and did their project for this unit, and I think it really helped them gain confidence that they were looking at the right set of notes and doing the right thing. I accidentally included a standard form to factored form problem that is not factorable, and although that led to a good reminder that not all quadratics are factorable, I did change it to vertex form on the file for next year.

By the end of this unit, my students for the most part felt really comfortable with the processes involved in factoring, completing the square, and multiplying out polynomials.

 

You can find the files for these pages here, in PDF and Publisher form.

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Learning, Growing, Laughing, Healing – A Year of Math Class

I recently wrote a post about my fifth year teaching. In that post, I talk about how my school building has come to feel like family. And right now, we’re down to 5 days left with that family for this year.

This time of year, I like to take one class period to pause, and give my students some time to reflect. I want them to take a minute to think about all the actual stuff they’ve learned this year, because I know they don’t realize in the day to day how much it adds up to. I want them to have time to remember all the times in our classroom together, their favorite and least favorite parts. I give this assignment under the guise of them helping me become a better teacher, and it definitely does give me lots of things to think about and consider every summer, but the real purpose of it is to give them a chance to be proud of themselves. Because they should be.

The past few years, I’ve made a post with some of the highlights from this assignment. You can read those posts here and here. Below, you can see this year’s highlights.

Math

  • I learned that vertex form can’t be turned into factored form without going to standard form first
  • Something that will always stick with me is my revitalized interest in math, and the feeling of accomplishment when I understand concepts I believed to be far beyond my comprehension
  • I understand slope much better than I did last year. Most of these concepts didn’t make sense to me before but you really helped me understand math better.
  • The thing that will probably stay with me is factored to standard and vertex to standard form. I could explain those things to other people because I love those things
  • [Graphing] polynomials with no calculator was my favorite!

Life Skills

  • I learned that there is no shame in asking questions, even if they seem obvious or repetitive
  • My notes will stay with me for the entire summer vacation so I won’t forget how to do math equations
  • I learned I’m good at organizing my notebook. I love my notebook so much and all the stickers I got.
  • Thought I’ve made multiple mistakes, but I asked for help when needed which I always do get help.

Community

  • My favorite part was all the jokes with the squad
  • I like that you give us house points to motivate us
  • I like most staff members, they are very nice. I like how Miss Mastalio is very helpful and explained my math problems better. There was never a dull moment at Mid City
  • I also like how the groups are arranged by Hogwarts characters [houses]. It makes me feel like I’m actually in a cool class and I’m proud to be a Ravenclaw!
  • I liked how Mid City does their fun stuff and actually wants/makes sure the students are happy
  • I liked the way you taught and how patient you were, and that you’d always help
  • Something that is going to stay with me this year is the great memories of this class and how much we laughed
  • I have also learned a lot of great 80’s music. Thanks [student name], [student name], and Mastalio for singing them
  • I loved calling Mastalio a vampire [note, this is a part of my Algebra 2 class’ intensely elaborate inside joke about my backstory, which involves me being a 2,000 year old vampire witch catwoman who had George Washington’s baby. Maybe just don’t ask about it]

Everything’s fine, I’m just crying

  • In all seriousness I really appreciate you, and I looked forward to your class every day. You’re the best
  • I learned that some teachers actually love me
  • I learned ANY ONE CAN BE A MATH PERSON
  • I like how calm, understanding, and accepting both this school and you have been I’ve never been more than average at math so it was nice to have a teacher and environment that was helpful and very motivated. I certainly feel like I’m much better at math than I would be without that.
  • [I learned] that anybody can be a great math magician

 

Overall, my students commented that they liked when we did poster projects instead of quizzes or tests, which is motivating me to work on different assessment styles for next year (I really want to experiment with group or partner assessments, or assessments where students get 5 minutes to talk with a partner before they start, etc, and with portfolio style assessments). A lot of them also commented that their least favorite practice assignments were when we just had a list of problems to solve, so I’ll also be working on replacing more of those with interactive activities like Question Stacks or card sorts or dice games.

Many of these students comment about things that seem like they should be a given in any classroom, but I know they’ve come from traumatic experiences with math classes before – classes where their teacher told them to their face that they’d never be successful, classes where they asked for help and did not get it, classes where instruction they missed was never explained to them. Reading what they wrote about feeling comfortable asking for help in my class, knowing they’ll always get it, understanding and making sense of notes…it makes me really proud of them for working to heal their math trauma. I read a twitter thread recently in which someone referred to their realization that their students DIDN’T have math phobia, they had math trauma. It’s a realization I’ve also come to. Someone, or several people, DID THIS TO THEM. They did not just start out afraid of math. There’s a reason they don’t like it, feel they can’t do it, won’t even try. It’s my job to try to start healing that. It’s a process that often can’t be completed in one year, but their words on these reflections provide evidence that it is happening, it is possible, and we just have to keep working to make math a positive experience for them.

 

Five Years.

FIVE years ago, at this time, I had recently accepted a job at a place called the Kimberly Center.

I was unsure of my decision. I had applied for jobs in various parts of Eastern/Central Iowa, knowing I didn’t want to move too incredibly far away from my parents (Iowa City) and sister (Davenport), but mostly just wanting a job. A classroom to teach in, finally.

I had been offered positions at a few other schools. I had laid awake on sleepless nights trying to decide, talked to my mom, my best friends, my cooperating teacher for student teaching. I had to decide: middle school? Freshmen at a big high school? Various 9-12 in a small town? Or, this “alternative school”.

I don’t think anyone else’s first choice for me was the alternative school (besides the principal who wanted to hire me). People told me it sounded really challenging, the words “scary” and “dangerous” were used. But I’ve always had a stubborn streak, and so I think in the end I chose it mostly to prove I could.

 

Fast forward to my first day of school – my first month, my first year even. There were so many sleepless nights. I was so overwhelmed. Also, I was terrible at asking for help. Poor Heather would come to my classroom after school and ask if I needed anything, I would say no, she would leave with a kind reminder that I could always ask her, and then I would go home and cry. I just didn’t know what I was doing, plain and simple.

My students had been through so much that I didn’t even know existed when I was their age. It took me most of that first year to really understand how their priorities worked and to shift from a whole ton of sympathy (feeling sorry for them and for myself) to true empathy.

 

In the past five years, I have learned more than I did in the 22 years that came before them. I have grown as a person into someone that I am really, truly proud of being.

In this school, now Mid City, I have found another home. I have found a family. I have found so, so much more than I ever thought I would walking through the doors for the first time five years ago.

My coworkers have become good friends – friends who get me and my band obsessions, my outrage at sexism in the sports world, my need for time away from people, my need for data and information. We have theme days and staff socials that turn into karaoke nights and we have endless, endless inside jokes.

My administration treats me with respect and importance – they make me feel like I am an expert in math education and take my input on changes and ideas. I know I can approach them if I disagree with something, or if I have an idea.

 

My students. Man, my students. I went from not understanding them at all, and having different goals for them than they did for themselves, to truly trying to work together to reach for whatever they think success looks like for them.

In the past five years, they have made me laugh every single day. From the things you can see browsing #thingsstudentssay on my twitter to more complicated and subtle inside jokes with different class periods.

They have made me cry, for all different reasons. Because they frustrate me. Because I hurt for them. Because they’ve lost loved ones or we have lost part of our Maverick Family. Because I am so incredibly, wonderfully proud of them.

They’ve done math that I know they never believed they could. Statements have come out of their mouth like, “putting quadratics in vertex form is fun!” and “I tried the challenge problem, got out Desmos and put some things in, played around with it for awhile, and I couldn’t get the y-intercepts to match. But can we talk about it later? Because I still want to know how it works.” – statements that they probably wouldn’t have known what they meant when the school year started, and if they had, would have laughed at someone else saying them. They’ve graphed quadratics by hand, gone from not understanding division to being able to complete the square (a certain student’s literal growth from this year), written paragraphs using statistics to compare athletes, won our school’s bracket competition using statistical analysis, and so much more.

They celebrate when we reach page 100 of our interactive notebooks. They take home papers with stickers to show their moms. 4 of my 5 class periods have said they want to do extra dot talks on the last day of school because they didn’t want this week to be their last Mental Math Monday of the year. They form identification with their Hogwarts Houses and do puzzles and challenges and put away my calculators and plug in my chromebooks to earn points for their House.

We’re a family. They’re my people. Some of them call me Mom and bring me dandelions, or cookies, or whatever they make in foods class. They find me in the morning to say good morning, in the halls to say they miss my class from last year. They ask what I’m teaching next year and if they can take that class. They tell me about crushes, girlfriends, boyfriends, breakups, nieces, nephews, college acceptances, trips, concerts, and more.

They push me to be better every single day. A better person and a better teacher. They aren’t scared to let me know when I slip into lowering expectations. They don’t listen to my instructions but they notice when I get frustrated about repeating them fifteen times after they start working. They ask how my day is going and they truly listen to my answer – way more than adults in my life do.

 

And they GRADUATE, something many of them never saw in their futures, they GRADUATE and they walk across that stage and then they become facebook friends and I watch them continue to learn and grow and say such heartfelt things about my classroom.

 

I have grown obsessed with the mathematicians from the Hidden Figures story in the past two years. My mom recently brought me an interview with Katherine Johnson from the paper. The entire thing resonated with me so strongly, but one line in particular stood out. She said, when asked about all the trials and everything she went through during her time at the space program, “I believed I was where I was supposed to be.”

I have days when I wake up and my first thought is “ugh, no, back to bed”. I have days where I still go home and cry from frustration or failure. There are days when I don’t think I can deal with a particular student anymore. “Do what you love and you’ll never have to work a day in your life” is a LIE, and don’t let anyone tell you differently. I work SO hard, many times harder than I should, because I care so much. It’s so hard to be a teacher, and even harder to work with at risk students.  I still love my job. It’s not just a place to go to get paid. It fulfills me and brings purpose and joy and heart to my life.

Five years have passed since I graduated college and accepted this job. Five years of making my classroom my own. Five years of joy, and pain, heartache and pride and laughter and learning and math.

Five years later, I believe I am where I am supposed to be. Here, at Mid City.

Algebra 2 Unit 8 Interactive Notebooks: Sketching Polynomial Graphs

This year I’ve committed to posting each unit of both my Algebra 1 and Algebra 2 INBs.

My district is moving to a standards based curriculum, and has identified priority standards for every course. These are the standards we are required to address and assess our students over, so they pretty much form our units.

You can find my Algebra 1 (year long class) INB posts here:

Unit 1 | Unit 2 | Unit 3 | Unit 4 | Unit 5 | Unit 6 | Unit 7

And my Algebra 2 INB posts here:

Unit 1 | Unit 2 | Unit 3 | Unit 4 | Unit 5 | Unit 6 | Unit 7

And finally, my posts from a 2nd go around I’m teaching of Algebra 1 here:

Unit 1 | Unit 2 | Unit 3 | Unit 4

 

Our 8th priority standard in Algebra 2 is A.APR.3:

Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

 

This is a no calculator allowed unit, which makes sense because the goal is to create a ROUGH graph based on the factored form of the function. My students actually found the freedom from calculators nice – as one of them put it, “It makes me feel like I’m not going to make it more complicated than it needs to be, because I know I shouldn’t need a calculator to do it.”

 

Here’s how I broke down this unit into skills (you can see from the index page picture that I changed the wording of one of our goals after I made it):

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Skill 1: I can identify zeros of a polynomial when it is given in factored form

When we had done our solving polynomials unit earlier in the year, my students expressed a lot of confusion regarding the words root, zero, solution, and x-intercept. I decided to address this for this unit – I actually only got really clear about which term should be used in which circumstance when I saw a graphic someone had posted on twitter last year, so I’m unsurprised that students were confused! I took extra care throughout this unit to use the correct terms with the correct situations and I think my students feel a lot more comfortable using each of them now.

After discussing those terms, we practiced finding the zeros of several factored functions, and then matching them to a graph with x-intercepts that made sense. My students found this easy, although we had to take a brief detour into the world of fractions since they couldn’t use a calculator to find a decimal equivalent. There were a lot of number lines and counting by the denominator that you don’t see in these notes, but I think it was a really good revisit of fraction concepts!

Skill 2: I can identify the total number of solutions, maximum number of extrema, and end behavior based on the degree of a polynomial function

This first page on information you can tell from a polynomial function could use some reformatting. I really liked the root types and the possibilities for what degree 3 solutions could look like on a graph, but I want to switch their places on the page. My students also kept forgetting the difference between finding the degree of a factored polynomial function and one in standard form, so I would add that at the top of the page.

The graph shape and end behavior page is the one that got used the most during this unit. Pretty much all of the other information got memorized pretty quickly, but this one was the page they kept referencing. We used Desmos to explore what would happen for each of the scenarios, with them choosing an even degree and a positive leading coefficient, then having another student choose an even degree and positive leading coefficient, etc., which was really helpful for them to see that this end behavior would hold true for ANY even degree and ANY positive leading coefficient. We also got to see some interesting graphs, like y=57x^100…

Skill 3: I can sketch a rough graph of a polynomial using its factored equation

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We’re putting it all together! I liked the organization of the information to identify here. The only struggle my students had was that the zeros don’t always end up in the order that the x-intercepts appear on the graph, so for the example on the front, they kept putting the double root on (4,0) instead of (-1,0) because that was the order it was listed in the equation. I might add a place for them to rearrange the x-intercepts and types in order to try to prevent this.

Skill 4: I can write a polynomial from given constraints

This is going backwards from what we had just been doing, so it went pretty well intuitively for the students. I always frame this section as them being the teachers and trying to come up with problems. For their assignment, I actually had them write functions, then find the roots and initial value and write them on an index card, and then they traded to see if they could get back to the original function! We mostly looked at having each factor having degree 1, but the last example I showed them how you could come up with alternatives by using other exponents on factors.

 

You can find the files for these notes here, in PDF and Publisher form.

Algebra 2 Unit 7 Interactive Notebooks: Probability

This year I’ve committed to posting each unit of both my Algebra 1 and Algebra 2 INBs.

My district is moving to a standards based curriculum, and has identified priority standards for every course. These are the standards we are required to address and assess our students over, so they pretty much form our units.

You can find my Algebra 1 (year long class) INB posts here:

Unit 1 | Unit 2 | Unit 3 | Unit 4 | Unit 5 | Unit 6 | Unit 7

And my Algebra 2 INB posts here:

Unit 1 | Unit 2 | Unit 3 | Unit 4 | Unit 5 | Unit 6

And finally, my posts from a 2nd go around I’m teaching of Algebra 1 here:

Unit 1 | Unit 2 | Unit 3 | Unit 4

The 7th standard I cover in Algebra 1 is not a priority standard, it is one of the “if you have time” standards. I chose to leave the last two priority standards until the last two units of the year, because then they would have covered them closer to taking their final. This standard is one that I really love, and was kind of bummed when it wasn’t chosen as a priority standard, so when I figured out I would have time to do an extra one, I jumped right to it.

S.CP.9: Use permutations and combinations to compute probabilities of compound events and solve problems.

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Skill 1: I can calculate experimental and theoretical probabilities

To kick off this unit, we play a game of BLOCKO! from Sarah Carter, who got it from Natalie Turbiville. I have a Katie Kubes 3D Cube Model set that our industrial tech teacher got from a conference that I never use for the actual 3D modeling but works perfectly for this purpose. As their posts state, you don’t tell the students the rules before they place their cubes the first time, and they get very frustrated when they realize how long it’s going to take for them to remove them all!

We then discussed the difference between theoretical and experimental probability, including us actually flipping a coin – aka one of my students finding ways to bounce it off the wall, ceiling, and tables… I got this notes page from Sarah Carter as well, but I cannot seem to find the post with her file to it at this moment! Then we filled out a chart of theoretical probabilities of rolling two dice (find this file in Sarah’s BLOCKO! post), calculated the probability of each outcome, and then the students placed their blocks again. They noticed that the game was much shorter this time, and made some good changes to their layouts! We tracked the experimental probability of each outcome throughout this game and then discussed the differences.

To close out our intro information, we defined sample space and discussed the fundamental counting principle. I am not entirely convinced that the information I found on possible Social Security numbers is accurate…and my students really wanted me to tell them my Social Security number…nope.

Skill 2: I can calculate permutations in appropriate situations

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To start off our discussion of permutations, I gave my students 5 minutes and the challenge to arrange the numbers 1,2,3, and 4 as many ways as they could. They got very competitive about this challenge and kept asking, “how many are there? Do I have them all?” which, of course, led perfectly into our discussion!

A few of my students got all 24 permutations, which was awesome. We looked at several other examples of permutations, and after we looked at permutations of my name, I had them each find the permutations of their own names. One of them asked if we could do the challenge again with the extra time we had left at the end of class, but with the numbers 1-5. Some of them started listing arrangements, but then someone calculated how many permutations there were for this, and they decided they did not have the patience to write out 120 different arrangements!

Skill 3: I can calculate combinations in appropriate situations

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We also started off this lesson with a challenge: how many different ways are there to choose two letters from A,B,C, and D? “This is easy!” They all exclaimed, thinking of our previous challenge…until I started walking around and pointing out to them that AB and BA were choosing the same letters. They ended up being very unsure about how many possibilities there were here:

We did a few examples of how to find combinations, and then looked at a set of scenarios where they had to decide if it was a permutation or combination needed. This always proves to be an interesting discussion, because my students ALWAYS overthink the situations. The last question on this note page brings them back into the realm of probability, so they can start to see how these skills are related.

Skill 4: I can use probability addition rules

I really like how I started this page with conceptual ideas of mutually exclusive and inclusive, before we got into the probabilities. Students connect pretty strongly with these examples and have good discussions, and this class even started coming up with their own examples when we finished these!

You may notice that in the Venn Diagram I used teachers from my own school…which I will have to change next year because one of them is leaving 😦 😦 😦 You would want to change this to teachers from your school if you use this. My students also called me out on making the numbers up and not using actual overlaps between our classes, so I should probably change the numbers for next time….maybe collect data from my students like I did for a page later in this unit!

Skill 5: I can identify the difference between independent and dependent events

To set up a discussion of independent and dependent events, I spent a lesson playing two games. Probability Bingo is from Sarah Carter – she details her journey of trying to color foam cubes in her post, and I have two large red foam cubes as part of a 3D objects set and did not want to even try coloring them, nor did I really want permanent writing on them in case I wanted to use them for something else later. I ended up changing her colors to shapes – heart, star, and circle – and cutting them out of printer paper and using a glue stick to stick them to the foam. They’ve now lasted two years of this lesson, but I can still peel the shapes off with no damage to the cubes if I want to! I think I want to change the circles to triangles or something next year, because my students keep thinking the circles are zeros when they try to list their possibilities and then getting confused. I’ve played this two ways – once where we play without looking at probabilities first, and once where we start off with the probabilities. The game was enjoyable both ways, so it depends on how much time you have.

We also play Egg Roulette. I’ve seen this on twitter several times where people play with Easter eggs that have confetti inside 3 of them…but I decided to go all in last year and it was GREAT. This is actually the only time all year I hardboil eggs because I don’t like them, so I’m always mildly concerned that I won’t do it right, but they’ve come out ok so far. You boil 9 out of a dozen eggs, and then mix them up in the carton. Bring in a big tupperware container and let your students take turns selecting an egg and smashing it into the tupperware, but BEFORE they smash it, calculate the probability that they will smash a raw one. If they smash a raw egg, they’re out!

Then we discuss the differences between these two activities and how the probabilities worked with in them, using that as a launching point to discuss independent and dependent events. We brainstorm more ideas of both types of event, and then do some sample calculations to compare the difference.

Skill 6: I can calculate conditional probabilities

We begin by conceptually thinking of conditional probabilities: if this thing is already true, what is the probability of this other thing?

Then we move into talking about the formula for calculating conditional probabilities. Every Friday, our opener question is a fun question to help me get to know my students better. One Friday before this lesson, I took the opportunity to use this question to collect some data: I asked my students from all my classes what their favorite fast food restaurant was, and if they’d eaten there in the last week. That’s where this data came from, which made it more interesting for my students to do calculations with, because then they were judging everyone’s fast food preferences.

Last, we used the formula to make calculations when just given probabilities, moving back into more algebraic manipulation.

 

You can find all the files from this post in links (if they are someone else’s) or here (if they’re mine), in Publisher and PDF form.